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Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, category theory is used in almost all areas of mathematics, and in some areas of computer science. In particular, many constructions of new mathematical objects from previous ones, that appear similarly in several contexts are conveniently expressed and unified in terms of categories. Examples include quotient spaces, direct products, completion, and duality.
Semantics is the study of reference, meaning, or truth. The term can be used to refer to subfields of several distinct disciplines, including philosophy, linguistics and computer science.
Where Mathematics Comes From: How the Embodied Mind Brings Mathematics into Being is a book by George Lakoff, a cognitive linguist, and Rafael E. Núñez, a psychologist. Published in 2000, WMCF seeks to found a cognitive science of mathematics, a theory of embodied mathematics based on conceptual metaphor.
George Philip Lakoff is an American cognitive linguist and philosopher, best known for his thesis that people's lives are significantly influenced by the conceptual metaphors they use to explain complex phenomena.
The philosophy of mathematics is the branch of philosophy that studies the assumptions, foundations, and implications of mathematics. It aims to understand the nature and methods of mathematics, and find out the place of mathematics in people's lives. The logical and structural nature of mathematics makes this branch of philosophy broad and unique.
In cognitive linguistics, conceptual metaphor, or cognitive metaphor, refers to the understanding of one idea, or conceptual domain, in terms of another. An example of this is the understanding of quantity in terms of directionality or the understanding of time in terms of money.
Cognitive linguistics is an interdisciplinary branch of linguistics, combining knowledge and research from cognitive science, cognitive psychology, neuropsychology and linguistics. Models and theoretical accounts of cognitive linguistics are considered as psychologically real, and research in cognitive linguistics aims to help understand cognition in general and is seen as a road into the human mind.
Cognitive science is the scientific study either of mind or of intelligence . Practically every formal introduction to cognitive science stresses that it is a highly interdisciplinary research area in which psychology, neuroscience, linguistics, philosophy, computer science, anthropology, and biology are its principal specialized or applied branches. Therefore, we may distinguish cognitive studies of either human or animal brains, the mind and the brain.
Saunders Mac Lane was an American mathematician who co-founded category theory with Samuel Eilenberg.
In cognitive linguistics, conceptual blending, also called conceptual integration or view application, is a theory of cognition developed by Gilles Fauconnier and Mark Turner. According to this theory, elements and vital relations from diverse scenarios are "blended" in a subconscious process, which is assumed to be ubiquitous to everyday thought and language. Much like memetics, it is an attempt to create a unitary account of the cultural transmission of ideas.
Construction grammar is a family of theories within the field of cognitive linguistics which posit that constructions, or learned pairings of linguistic patterns with meanings, are the fundamental building blocks of human language. Constructions include words, morphemes, fixed expressions and idioms, and abstract grammatical rules such as the passive voice or the ditransitive. Any linguistic pattern is considered to be a construction as long as some aspect of its form or its meaning cannot be predicted from its component parts, or from other constructions that are recognized to exist. In construction grammar, every utterance is understood to be a combination of multiple different constructions, which together specify its precise meaning and form.
In mathematical logic, the Cantor–Dedekind axiom is the thesis that the real numbers are order-isomorphic to the linear continuum of geometry. In other words, the axiom states that there is a one-to-one correspondence between real numbers and points on a line.
Raymond Louis Wilder was an American mathematician, who specialized in topology and gradually acquired philosophical and anthropological interests.
Reuben Hersh was an American mathematician and academic, best known for his writings on the nature, practice, and social impact of mathematics. Although he was generally known as Reuben Hersh, late in life he sometimes used the name Reuben Laznovsky in recognition of his father's ancestral family name. His work challenges and complements mainstream philosophy of mathematics.
Embodied cognitive science is an interdisciplinary field of research, the aim of which is to explain the mechanisms underlying intelligent behavior. It comprises three main methodologies: the modeling of psychological and biological systems in a holistic manner that considers the mind and body as a single entity; the formation of a common set of general principles of intelligent behavior; and the experimental use of robotic agents in controlled environments.
Experientialism is a philosophical view which states that there is no "purely rational" detached God's-eye view of the world which is external to human thought. It was first developed by George Lakoff and Mark Johnson in Metaphors We Live By. Experientialism is especially a response to the objectivist tradition of transcendental truth most prominently formulated by Immanuel Kant which still requires a commitment to what Lakoff and Johnson call "basic realism". Most importantly, this involves acknowledging the existence of a mind-independent external world and the possibility of stable knowledge of that external world. In Women, Fire and Dangerous Things, Lakoff expands on the foundations of experientialism with research into the nature of categories.
Embodied cognition is the theory that many features of cognition, whether human or otherwise, are shaped by aspects of an organism's entire body. Sensory and motor systems are seen as fundamentally integrated with cognitive processing. The cognitive features include high-level mental constructs and performance on various cognitive tasks. The bodily aspects involve the motor system, the perceptual system, the bodily interactions with the environment (situatedness), and the assumptions about the world built into the organism's functional structure.
In linguistics, the term formalism is used in a variety of meanings which relate to formal linguistics in different ways. In common usage, it is merely synonymous with a grammatical model or a syntactic model: a method for analyzing sentence structures. Such formalisms include different methodologies of generative grammar which are especially designed to produce grammatically correct strings of words; or the likes of Functional Discourse Grammar which builds on predicate logic.
Theory of language is a topic from philosophy of language and theoretical linguistics. It has the goal of answering the questions "What is language?"; "Why do languages have the properties they have?"; or "What is the origin of language?".
References
- Gårding, Lars, 1977. Encounter with Mathematics. Springer-Verlag.
- Reuben Hersh, 1997. What Is Mathematics, Really? Oxford Univ. Press.
- George Lakoff and Rafael E. Núñez, 2000. Where Mathematics Comes From . Basic Books.
- Mac Lane, Saunders (1986). Mathematics, Form and Function. Springer-Verlag. ISBN 0-387-96217-4.
- Leslie White, 1947, "The Locus of Mathematical Reality: An Anthropological Footnote," Philosophy of Science 14: 289-303. Reprinted in Hersh, R., ed., 2006. 18 Unconventional Essays on the Nature of Mathematics. Springer: 304–19.